A187060 -id:A187060 - cp's OEIS frontend

A190800 Primes p such that the polynomial x^2 + x + p generates only primes for x = 1..8.

11, 17, 41, 844427, 51448361, 51867197, 85776137, 86966771, 93685301, 97122197, 107599757, 113575727, 118136267, 122983031, 180078317, 232728647, 316973621, 483040757, 564537761, 749930717, 840472307, 901288517, 960959381, 1278189947, 1559839991, 1696818647, 1761702947, 1829187287, 2251028567

Offset: 1

Zak Seidov, Jun 02 2011

nonn

Subsequence of A187060: a(1..10)=A187060(1,2,3,7,18,19,26,28,31,32).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A187060.

Select[Prime[Range[110*10^6]],AllTrue[Table[x^2+x+#,{x,8}],PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jul 03 2020 *)

is(n)=for(x=0,8, if(!isprime(x^2+x+n), return(0))); 1 \\ Charles R Greathouse IV, Sep 14 2015

A190838 Initial primes of 8 consecutive primes with 7 consecutive gaps 2, 4, 6, 8, 10, 12, 14.

Original entry on oeis.org

128981, 21456047, 34864211, 51867197, 55793951, 69726647, 113575727, 180078317, 207664397, 232728647, 342241967, 382427027, 382533311, 470463011, 558791327, 591360851, 603413801, 749930717, 838115711, 926976431, 965761397, 1007421251, 1109867567, 1278189947

Offset: 1

Zak Seidov, May 21 2011

nonn

a(1) = 128981 = A190819(1), a(2) = 21456047 = A190819(14).

a(n) + 56 is the greatest term in the sequence of 8 consecutive primes with 7 consecutive gaps 2, 4, 6, 8, 10, 12, 14. - Muniru A Asiru, Aug 10 2017

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Index entries for primes, gaps between

Subsequence of A190819.
Subsequence of A187060. - Michel Marcus, Aug 10 2017
Cf. A078847, A190814, A190817.

N:=10^8:  # to get all terms <= N.
Primes:=select(isprime,[seq(i,i=3..N+56,2)]):
Primes[select(t->[Primes[t+1]-Primes[t], Primes[t+2]-Primes[t+1],
  Primes[t+3]-Primes[t+2], Primes[t+4]-Primes[t+3], Primes[t+5]-
  Primes[t+4], Primes[t+6]-Primes[t+5] , Primes[t+7]-Primes[t+6] ]=
[2,4,6,8,10,12,14], [$1..nops(Primes)-7])]; # Muniru A Asiru, Aug 04 2017

Transpose[Select[Partition[Prime[Range[65000000]],8,1],Differences[#] =={2,4,6,8,10,12,14}&]][[1]] (* Harvey P. Dale, May 10 2014 *)

list(lim)=my(v=List(),p=128981,t); forprime(q=p+2,lim+56, if(q-p-t==2, t+=2; if(t==14, listput(v, q-56); t=0), t=0); p=q); Vec(v) \\ Charles R Greathouse IV, Aug 10 2017

Additional cross references from Harvey P. Dale, May 10 2014

A191456 Primes p such that the polynomial x^2+x+p generates only primes for x=1..9.

Original entry on oeis.org

11, 17, 41, 844427, 51448361, 86966771, 122983031, 180078317, 960959381, 1278189947, 1761702947, 1829187287, 2426256797, 2911675511, 3013107257, 4778888351, 5221343711

Offset: 1

Zak Seidov, Jun 02 2011

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..2713

Generates primes for x=1..k: A001359 (1), A022004 (2), A172454 (3), A187057 (4), A187058 (5), A144051 (6), A187060 (7), A190800 (8), this sequence (9), A191457 (10), A191458 (11), A253592 (12), A253605 (13). Each is by definition a subsequence of preceding sequences.
Subsequence such that x=10 gives a composite number: A211238.
Cf. A160548, A164926.

is(n)=for(x=0,9, if(!isprime(x^2+x+n), return(0))); 1 \\ Charles R Greathouse IV, Sep 14 2015

A187058 Primes p such that the polynomial x^2 + x + p generates only primes for x = 1..5.

Original entry on oeis.org

11, 17, 41, 1277, 1607, 3527, 13901, 21557, 26681, 28277, 31247, 33617, 55661, 68897, 97367, 113147, 128981, 166841, 195731, 221717, 347981, 348431, 354371, 416387, 421697, 506327, 548831, 566537, 665111, 844427, 929057, 954257

Offset: 1

Jonathan Vos Post, Mar 03 2011

nonn

From Weber, p. 15.
This sequence is infinite, assuming Dickson's conjecture.
All terms = {11, 17} mod 30. - Zak Seidov, May 07 2011

			a(2) = 17 because x^2 + x + 17 generates 0^2 + 0 + 17 = 17; 1^2 + 1 + 17 = 19; 2^2 + 2 + 17 = 23; 3^2 + 3 + 17 = 29; 4^2 + 4 + 17 = 37; and 5^2 + 5 + 17 = 47, all primes.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000, replacing a b-file from Zak Seidov.
H. J. Weber, Regularities of Twin, Triplet and Multiplet Prime Numbers, arXiv:1103.0447 [math.NT], 2011-2012.

Cf. A144051, A187057, A187060.

okQ[n_] := And @@ PrimeQ[Table[i^2 + i + n, {i, 0, 5}]]; Select[Range[10000], okQ] (* T. D. Noe, Mar 03 2011 *)
Select[Prime[Range[76000]],AllTrue[#+{2,6,12,20,30},PrimeQ]&] (* Harvey P. Dale, Apr 12 2025 *)

forprime(p=9,1e6,if((p%30==11 || p%30==17) && isprime(p+2) && isprime(p+6) && isprime(p+12) && isprime(p+20) && isprime(p+30), print1(p", "))) \\ Charles R Greathouse IV, May 08 2011

A191458 Primes p such that the polynomial x^2+x+p generates only primes for x=1..11.

Original entry on oeis.org

17, 41, 1761702947, 8776320587, 10102729577, 11085833111, 177558051107, 273373448057, 473787509537, 557149355507, 715464238661, 1359854730821, 2131528031441, 2170341748697, 2236159108277, 2308235320997, 2751203698151, 3247566894821, 3288002848511, 3424305123047

Offset: 1

Zak Seidov, Jun 02 2011

nonn

Subsequence of A191457.

The sequence is infinite under Dickson's conjecture. - Charles R Greathouse IV, Oct 11 2011

Charles R Greathouse IV and Zak Seidov, Table of n, a(n) for n = 1..79

Cf. A187060, A190800, A191456, A191457.

is(n)=for(x=0,11, if(!isprime(x^2+x+n), return(0))); 1 \\ Charles R Greathouse IV, Sep 14 2015

a(12)-a(20) from Charles R Greathouse IV, Oct 17 2011

A191457 Primes p such that the polynomial x^2+x+p generates only primes for x=1..10.

Original entry on oeis.org

17, 41, 180078317, 1278189947, 1761702947, 1829187287, 5862143447, 6369321857, 7226006861, 8776320587, 10102729577, 11085833111, 12412643261, 50626299797, 53039299211, 72355485857, 74621287901, 76233413141, 81948881447, 115826556611, 129077263697

Offset: 1

Zak Seidov, Jun 02 2011

nonn

Dickson's conjecture implies that this sequence is infinite. [Charles R Greathouse IV, Jun 03 2011]

Zak Seidov, Table of n, a(n) for n = 1..464

Subsequence of A191456. Cf. A187060, A190800.

isokp(p) = {for (n=1, 10, if (! isprime(subst(x^2+x+p, x, n)), return (0));); 1;}
lista(nn) = {forprime (p=1, nn, if (isokp(p), print1(p, ", ")); ); } \\ Michel Marcus, Jan 05 2015

A273595 Least q > 0 such that min { x >= 0 | q + prime(n)*x + x^2 is composite } is a (local) maximum, cf. A273756 & A273770.

Original entry on oeis.org

43, 47, 53, 71, 83, 113, 131, 173, 251, 281, 383, 461, 503, 593, 743, 73361, 73421, 3071069, 15949847, 76553693, 2204597, 1842719, 246407807, 986578883, 73975907, 4069235123, 1244414939, 25213427, 656856899, 30641069183, 8221946477, 41730358853, 10066886927, 285340609997, 6232338461

Offset: 2

M. F. Hasler, May 26 2016

nonn

This is a subsequence of A273756 which considers all odd numbers (2n+1) instead of only prime(n) as coefficients of the linear term.
All terms are necessarily prime, since this is necessary and sufficient to get a prime for x = 0.
The respective minima (= number of consecutive primes for x = 0, 1, 2, ...) are given in A273597.
It has been pointed out by Don Reble that the prime k-tuple conjecture predicts infinitely long sequences of primes of the given form, therefore we consider the "local" maxima, for q below some appropriate (large) limit: see sequences A273756 & A273770 for further details. - M. F. Hasler, Feb 17 2020

Eric Weisstein's World of Mathematics, Prime-Generating Polynomial.
Index to entries related to primes produced by polynomials.

Cf. A273756, A273770.

Cf. also A002837 (n such that n^2-n+41 is prime), A007634 (n such that n^2+n+41 is composite), A005846 (primes of form n^2+n+41), A097823, A144051, A187057 .. A187060, A190800, A191456 ff.

A273595(n)=A273756(prime(n)\2) \\ changed Feb 17 2020

a(n) = A273756((prime(n) - 1)/2). - M. F. Hasler, Feb 17 2020

Edited and extended using A273756(0..100) due to Don Reble, by M. F. Hasler, Feb 17 2020

A273756 Least p for which min { x >= 0 | p + (2n+1)*x + x^2 is composite } reaches the (local) maximum given in A273770.

Original entry on oeis.org

41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, 197, 223, 251, 281, 313, 347, 383, 421, 461, 503, 547, 593, 641, 691, 743, 797, 73303, 73361, 73421, 73483, 3443897, 3071069, 3071137, 15949847, 76553693, 365462323, 365462399, 2204597, 9721, 1842719, 246407633, 246407719, 246407807, 246407897, 246407989

Offset: 0

M. F. Hasler, May 26 2016

nonn

All terms are prime, since this is necessary and sufficient to get a prime for x = 0.
The values given in A273770 are the number of consecutive primes obtained for x = 0, 1, 2, ....
Sequence A273595 is the subsequence of terms for which 2n+1 is prime.
For even coefficients of the linear term, the answer would always be q=2, the only choice that yields a prime for x=0 and also for x=1 if (coefficient of the linear term)+3 is prime.
The initial term a(n=0) = 41 corresponds to Euler's famous prime-generating polynomial 41+x+x^2. Some subsequent terms are equal to the primes this polynomial takes for x=1,2,3,.... This stems from the fact that adding 2 to the coefficient of the linear term is equivalent to shifting the x-variable by 1. Since here we require x >= 0, we find a reduced subset of the previous sequence of primes, missing the first one, starting with q equal to the second one. (It is known that there is no better prime-generating polynomial of this form than Euler's, see the MathWorld page and A014556. "Better" means a larger p producing p-1 primes in a row. However, the prime k-tuple conjecture suggests that there should be arbitrarily long runs of primes of this form (for much larger p), i.e., longer than 41, but certainly much less than the respective p. Therefore we speak of local maxima.)

Don Reble, Table of n, a(n) for n = 0..100
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial
Index to the OEIS, Entries related to primes produced by polynomials.

Cf. A273595, A273770.
Cf. also A002837 (n such that n^2-n+41 is prime), A007634 (n such that n^2+n+41 is composite), A005846 (primes of form n^2+n+41), A097823, A144051, A187057 ... A187060, A190800, A191456 ff.
The first line of data coincides with that of A202018, A107448, A155884 (and also A140755, A142719, except for some initial terms), which are all related.

A273756(n,p=2*n+1,L=10^(5+n\10),m=0,Q)={forprime(q=1,L, for(x=1,oo, ispseudoprime(q+p*x+x^2)&& next; x>m&& [Q=q,m=x]; break));Q}

Edited, following a remark by Don Reble, by M. F. Hasler, Jan 23 2018

a(27) corrected and more terms from Don Reble, Feb 15 2018

A273770 Max { min { x >= 0 | p + (2n+1)x + x^2 is composite }, p < 10^(5+n/10) }.

Original entry on oeis.org

40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 13, 12, 11, 10, 11, 12, 11, 12, 12, 13, 12, 12, 13, 16, 17, 16, 15, 14, 13, 13, 12, 11, 12, 13, 13, 14, 13, 13, 13, 12, 13, 14, 13, 14, 15, 14, 14, 13, 14, 14, 13

Offset: 0

M. F. Hasler, May 26 2016

nonn

The values for p are given in A273756 which is the main entry, see there for further information and (cross)references.
From the initial values, the sequence seems strictly decreasing, with a(n) = 40-n, however, this property does not persist beyond a(27) = 13.
The upper limit on p ensures that we have a well-defined sequence: The prime k-tuple conjecture predicts existence of arbitrarily long sequences of primes of the given form, and thus unbounded minimal value of x. However, the corresponding prime tuples are expected to appear for much larger values of p. The given limit should be understood as "below the first/next such prime tuple", and in general the values a(n) should not change if that limit would be increased by some orders of magnitude. There might be counterexamples, which would be interesting. The given limit was chosen for lack of a more natural expression, and is relatively small. It could be replaced by a more appropriate function of n if a proposal is available, which should not affect the values given so far. - M. F. Hasler, Jan 22 2018, edited Feb 17 2020

M. F. Hasler, Table of n, a(n) for n = 0..100 (using A273756(0..100) from Don Reble), Feb 17 2020
Index to sequences related to primes produced by polynomials.

Cf. A273595, A273756.

Cf. also A002837, A007634, A005846, A097823, A144051, A187057 ... A187060, A190800, A191456 ff.

{A273770(n, p=2*n+1, L=10^(5+n/10), m)=forprime(q=1,L, for(x=1,oo, ispseudoprime(q+p*x+x^2) || (x>m && !m=x) || break));m}

a(n) = 40 - n for 0 <= n <= 27.

Corrected and extended by Don Reble, Feb 15 2018

A247949 Primes p such that the polynomial k^4 + k^3 + k^2 + k + p yields only primes for k = 0...5.

Original entry on oeis.org

7, 43, 79, 457, 877, 967, 1093, 2437, 2683, 3187, 5077, 5923, 7933, 8233, 11923, 12889, 15787, 17389, 19993, 31543, 41113, 41617, 42457, 71359, 77863, 80683, 91393, 101719, 102643, 105967, 107347, 120163, 129733, 137593, 151783, 170263, 175723, 197569, 210127

Offset: 1

K. D. Bajpai, Jan 11 2015

nonn

All terms == 1 mod 6. - Robert Israel, Jan 11 2015

			a(1) = 7:
0^4 + 0^3 + 0^2 + 0 + 7 = 7;
1^4 + 1^3 + 1^2 + 1 + 7 = 11;
2^4 + 2^3 + 2^2 + 2 + 7 = 37;
3^4 + 3^3 + 3^2 + 3 + 7 = 127;
4^4 + 4^3 + 4^2 + 4 + 7 = 347;
5^4 + 5^3 + 5^2 + 5 + 7 = 787;
all six are primes.

K. D. Bajpai, Table of n, a(n) for n = 1..6730

Cf. A144051, A187057, A187058, A187060, A190800, A191456, A191457, A191458.

select(p -> andmap(isprime, [p, p+4, p+30, p+120, p+340, p+780]), [seq(6*i+1, i=1..10^5)]); # Robert Israel, Jan 11 2015

Select[f=k^4 + k^3 + k^2 + k; k = {0, 1, 2, 3, 4, 5}; Prime[Range[2000000]], And @@ PrimeQ[#+f] &]
Select[Prime[Range[20000]],AllTrue[#+{4,30,120,340,780},PrimeQ]&] (* Harvey P. Dale, Dec 24 2023 *)

forprime(p=1, 500000, if( isprime(p+0)& isprime(p+4)& isprime(p+30)& isprime(p+120)& isprime(p+340)& isprime(p+780), print1(p,", ")))

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A190800 Primes p such that the polynomial x^2 + x + p generates only primes for x = 1..8.

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A190838 Initial primes of 8 consecutive primes with 7 consecutive gaps 2, 4, 6, 8, 10, 12, 14.

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A191456 Primes p such that the polynomial x^2+x+p generates only primes for x=1..9.

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A187058 Primes p such that the polynomial x^2 + x + p generates only primes for x = 1..5.

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A191458 Primes p such that the polynomial x^2+x+p generates only primes for x=1..11.

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A191457 Primes p such that the polynomial x^2+x+p generates only primes for x=1..10.

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A273595 Least q > 0 such that min { x >= 0 | q + prime(n)*x + x^2 is composite } is a (local) maximum, cf. A273756 & A273770.

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A273756 Least p for which min { x >= 0 | p + (2n+1)*x + x^2 is composite } reaches the (local) maximum given in A273770.

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A273770 Max { min { x >= 0 | p + (2n+1)x + x^2 is composite }, p < 10^(5+n/10) }.